Properties

Label 6003.2.a.j
Level $6003$
Weight $2$
Character orbit 6003.a
Self dual yes
Analytic conductor $47.934$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6003.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(47.9341963334\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 3 x^{6} - 5 x^{5} + 18 x^{4} + 4 x^{3} - 26 x^{2} + x + 8\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2001)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( \beta_{1} + \beta_{2} ) q^{4} + \beta_{4} q^{5} + ( -1 - \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} ) q^{7} + ( 1 + \beta_{2} + \beta_{3} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( \beta_{1} + \beta_{2} ) q^{4} + \beta_{4} q^{5} + ( -1 - \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} ) q^{7} + ( 1 + \beta_{2} + \beta_{3} ) q^{8} + ( 1 + \beta_{3} + \beta_{5} ) q^{10} + ( 1 - \beta_{1} - \beta_{3} - \beta_{4} ) q^{11} + ( -3 + \beta_{1} - \beta_{6} ) q^{13} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} ) q^{14} + ( -1 + \beta_{3} + \beta_{4} ) q^{16} + ( 1 - \beta_{2} - \beta_{4} - \beta_{5} ) q^{17} + ( -\beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{19} + ( \beta_{2} + \beta_{6} ) q^{20} + ( -3 - 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{22} - q^{23} + ( -\beta_{1} + \beta_{3} + 2 \beta_{5} - \beta_{6} ) q^{25} + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{4} - 2 \beta_{6} ) q^{26} + ( -\beta_{1} - \beta_{2} + \beta_{5} ) q^{28} + q^{29} + ( -3 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} ) q^{31} + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{32} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{34} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{35} + ( -5 + 3 \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{37} + ( -1 - 2 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{38} + ( -3 + \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{40} + ( 1 - \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{6} ) q^{41} + ( \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{43} + ( -3 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{6} ) q^{44} -\beta_{1} q^{46} + ( 4 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{47} + ( -2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{49} + ( -2 - 3 \beta_{1} - \beta_{2} + 2 \beta_{4} ) q^{50} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{52} + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{53} + ( -4 + \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{55} + ( -3 - \beta_{1} + \beta_{3} + 3 \beta_{4} - \beta_{6} ) q^{56} + \beta_{1} q^{58} + ( 2 \beta_{1} + \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} ) q^{59} + ( -3 - \beta_{1} + \beta_{3} - 2 \beta_{4} ) q^{61} + ( -2 - 3 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{5} - 2 \beta_{6} ) q^{62} + ( 1 - 3 \beta_{1} - 4 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{64} + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{65} + ( 3 - 3 \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{67} + ( -2 + 3 \beta_{1} - 2 \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{68} + ( -5 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{70} + ( 1 + \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{71} + ( 3 + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{73} + ( -2 - \beta_{1} + \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{74} + ( -3 - 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{76} + ( -2 + 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{77} + ( -1 - \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{6} ) q^{79} + ( 3 - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{80} + ( -3 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} ) q^{82} + ( -3 - \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 5 \beta_{6} ) q^{83} + ( -4 - \beta_{2} - 2 \beta_{5} + \beta_{6} ) q^{85} + ( 1 - 2 \beta_{1} + \beta_{3} + 5 \beta_{4} + \beta_{5} + 6 \beta_{6} ) q^{86} + ( 1 - 3 \beta_{1} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} ) q^{88} + ( -3 + 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{5} - \beta_{6} ) q^{89} + ( 3 - 2 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{91} + ( -\beta_{1} - \beta_{2} ) q^{92} + ( -4 + 3 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 5 \beta_{4} - \beta_{5} - 4 \beta_{6} ) q^{94} + ( 4 - 2 \beta_{2} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{95} + ( 3 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{5} ) q^{97} + ( -3 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} - 2 \beta_{5} + \beta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 3q^{2} + 5q^{4} + 3q^{5} - 5q^{7} + 6q^{8} + O(q^{10}) \) \( 7q + 3q^{2} + 5q^{4} + 3q^{5} - 5q^{7} + 6q^{8} + 3q^{10} + 4q^{11} - 18q^{13} + 2q^{14} - 7q^{16} + 3q^{17} - 4q^{19} + 2q^{20} - 26q^{22} - 7q^{23} - 8q^{25} + 7q^{26} - 6q^{28} + 7q^{29} - 22q^{31} - 5q^{32} + 9q^{34} - 3q^{35} - 25q^{37} - 14q^{38} - 10q^{40} + 13q^{41} - 2q^{43} - 4q^{44} - 3q^{46} + 25q^{47} - 8q^{49} - 19q^{50} - 12q^{52} + 5q^{53} - 15q^{55} - 18q^{56} + 3q^{58} - 11q^{59} - 33q^{61} - 28q^{62} - 14q^{64} + 2q^{65} + 8q^{67} - 12q^{68} - 22q^{70} + 6q^{71} + 15q^{73} - 34q^{74} - 28q^{76} + q^{77} - 15q^{79} + 12q^{80} - 14q^{82} - 21q^{83} - 28q^{85} + 12q^{86} - 13q^{88} - 8q^{89} + 6q^{91} - 5q^{92} - 35q^{94} + 25q^{95} + 13q^{97} - q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 3 x^{6} - 5 x^{5} + 18 x^{4} + 4 x^{3} - 26 x^{2} + x + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 3 \nu + 1 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - \nu^{3} - 5 \nu^{2} + 3 \nu + 4 \)
\(\beta_{5}\)\(=\)\( \nu^{5} - \nu^{4} - 6 \nu^{3} + 4 \nu^{2} + 7 \nu - 2 \)
\(\beta_{6}\)\(=\)\( \nu^{6} - \nu^{5} - 7 \nu^{4} + 5 \nu^{3} + 13 \nu^{2} - 5 \nu - 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 4 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{4} + \beta_{3} + 6 \beta_{2} + 6 \beta_{1} + 7\)
\(\nu^{5}\)\(=\)\(\beta_{5} + \beta_{4} + 7 \beta_{3} + 8 \beta_{2} + 19 \beta_{1} + 7\)
\(\nu^{6}\)\(=\)\(\beta_{6} + \beta_{5} + 8 \beta_{4} + 9 \beta_{3} + 32 \beta_{2} + 33 \beta_{1} + 31\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.98973
−1.21072
−0.586186
0.743347
1.44008
2.22973
2.37349
−1.98973 0 1.95904 1.78703 0 −0.840964 0.0814996 0 −3.55572
1.2 −1.21072 0 −0.534159 −3.03795 0 −3.40847 3.06816 0 3.67810
1.3 −0.586186 0 −1.65639 0.842866 0 1.23837 2.14332 0 −0.494076
1.4 0.743347 0 −1.44743 3.36180 0 −2.98340 −2.56264 0 2.49898
1.5 1.44008 0 0.0738306 −0.734622 0 3.73618 −2.77384 0 −1.05791
1.6 2.22973 0 2.97167 −0.537118 0 −1.88241 2.16657 0 −1.19762
1.7 2.37349 0 3.63344 1.31800 0 −0.859291 3.87694 0 3.12824
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(23\) \(1\)
\(29\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6003.2.a.j 7
3.b odd 2 1 2001.2.a.i 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.2.a.i 7 3.b odd 2 1
6003.2.a.j 7 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):

\( T_{2}^{7} - 3 T_{2}^{6} - 5 T_{2}^{5} + 18 T_{2}^{4} + 4 T_{2}^{3} - 26 T_{2}^{2} + T_{2} + 8 \)
\( T_{5}^{7} - 3 T_{5}^{6} - 9 T_{5}^{5} + 30 T_{5}^{4} - 5 T_{5}^{3} - 29 T_{5}^{2} + 6 T_{5} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 8 + T - 26 T^{2} + 4 T^{3} + 18 T^{4} - 5 T^{5} - 3 T^{6} + T^{7} \)
$3$ \( T^{7} \)
$5$ \( 8 + 6 T - 29 T^{2} - 5 T^{3} + 30 T^{4} - 9 T^{5} - 3 T^{6} + T^{7} \)
$7$ \( 64 + 156 T + 63 T^{2} - 93 T^{3} - 75 T^{4} - 8 T^{5} + 5 T^{6} + T^{7} \)
$11$ \( -776 - 1234 T - 251 T^{2} + 340 T^{3} + 87 T^{4} - 34 T^{5} - 4 T^{6} + T^{7} \)
$13$ \( 598 - 1027 T - 1054 T^{2} + 70 T^{3} + 317 T^{4} + 119 T^{5} + 18 T^{6} + T^{7} \)
$17$ \( 3088 - 1035 T - 1166 T^{2} + 388 T^{3} + 118 T^{4} - 39 T^{5} - 3 T^{6} + T^{7} \)
$19$ \( 2 + 21 T - 24 T^{2} - 273 T^{3} - 206 T^{4} - 37 T^{5} + 4 T^{6} + T^{7} \)
$23$ \( ( 1 + T )^{7} \)
$29$ \( ( -1 + T )^{7} \)
$31$ \( -16 - 134 T + 305 T^{2} + 802 T^{3} + 563 T^{4} + 168 T^{5} + 22 T^{6} + T^{7} \)
$37$ \( 151376 + 24287 T - 41305 T^{2} - 15769 T^{3} - 1411 T^{4} + 124 T^{5} + 25 T^{6} + T^{7} \)
$41$ \( -5732 + 11868 T - 5407 T^{2} - 547 T^{3} + 560 T^{4} - 21 T^{5} - 13 T^{6} + T^{7} \)
$43$ \( 2258 - 13069 T + 2914 T^{2} + 2417 T^{3} - 337 T^{4} - 134 T^{5} + 2 T^{6} + T^{7} \)
$47$ \( 105656 - 192742 T + 112001 T^{2} - 26420 T^{3} + 2169 T^{4} + 103 T^{5} - 25 T^{6} + T^{7} \)
$53$ \( 25268 - 21392 T - 31767 T^{2} + 9375 T^{3} + 658 T^{4} - 209 T^{5} - 5 T^{6} + T^{7} \)
$59$ \( 7552 - 59936 T + 93488 T^{2} + 11040 T^{3} - 2060 T^{4} - 210 T^{5} + 11 T^{6} + T^{7} \)
$61$ \( -153712 - 177502 T - 67029 T^{2} - 7465 T^{3} + 1163 T^{4} + 370 T^{5} + 33 T^{6} + T^{7} \)
$67$ \( -1070848 - 448844 T + 29863 T^{2} + 26061 T^{3} + 925 T^{4} - 287 T^{5} - 8 T^{6} + T^{7} \)
$71$ \( -78872 + 114117 T - 52936 T^{2} + 6220 T^{3} + 1209 T^{4} - 195 T^{5} - 6 T^{6} + T^{7} \)
$73$ \( -6928 + 14694 T - 7497 T^{2} - 1536 T^{3} + 1355 T^{4} - 73 T^{5} - 15 T^{6} + T^{7} \)
$79$ \( 142 - 591 T + 677 T^{2} - 51 T^{3} - 189 T^{4} + 12 T^{5} + 15 T^{6} + T^{7} \)
$83$ \( 1179352 + 1390690 T + 275481 T^{2} - 12993 T^{3} - 5640 T^{4} - 173 T^{5} + 21 T^{6} + T^{7} \)
$89$ \( -12688 + 11973 T + 39106 T^{2} + 3856 T^{3} - 1669 T^{4} - 191 T^{5} + 8 T^{6} + T^{7} \)
$97$ \( 457432 - 57666 T - 50171 T^{2} + 5355 T^{3} + 1545 T^{4} - 124 T^{5} - 13 T^{6} + T^{7} \)
show more
show less